2 Expression for Discontinuity Δ in Terms of S System Eigenvalues
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7.2 Expression for Discontinuity
in Terms of S System Eigenvalues
237
Since the external and Hartree potential energies also vanish in this limit we then
have
(N +ω)
(r) = 0.
(7.26)
lim vxc
r→∞
With this result, the S system orbital densities in general decay asymptotically as
e−2(−2 i )
1/2
|φi (r)|2
∼
r →∞
r
.
(7.27)
+ω)
The highest occupied fractionally charged orbital φ(N
N +1 (r) has the slowest decay
so that the asymptotic structure of the (N + ω)-electron system density is
ρ(N +ω) (r)
∼
r →∞
(N +ω) 1/2
r
N +1 )
e−2(−2
.
(7.28)
On the other hand, the asymptotic decay of the N - and (N + 1)-electron system
densities from (2.163) is
1/2
∼
e−2(2I N ) r ,
(7.29)
ρ(N ) (r) r →∞
and
ρ(N +1) (r)
e−2(2I N +1 )
1/2
∼
r →∞
r
,
(7.30)
where I N and I N +1 are the ionization energies for the N - and (N + 1)-electron
systems, respectively. Because the ionization energy for an (N + 1)-electron system
is smaller than that of an N -electron system we have
I N +1 < I N ,
(7.31)
and the fact (see (7.12)) that ρ(N +ω) (r) is a linear combination of ρ(N ) (r) and
ρ(N +1) (r), we have
ρ(N +ω)
∼
r →∞
ρ(N +1) (r)
e−2(2I N +1 )
1/2
∼
r →∞
r
.
(7.32)
A comparison of (7.32) with (7.28) leads to
(N +ω)
N +1
= −I N +1 =
(N +1)
N +1 ,
(7.33)
where the second equality is a consequence of the fact that the highest occupied
eigenvalue of the S system is minus the ionization potential (see Sect. 3.4.8). Equation
(7.33) also shows that the highest occupied eigenvalue is independent of the fractional
charge ω.
In order to show that vs(N +ω) (r) differs from vs(N ) (r) as ω approaches zero from
above, let us consider a radius R(ω) such that
ωρ(N +1) (R(ω)) = (1 − ω)ρ(N ) (R(ω)),
(7.34)
238
7 Quantal Density Functional Theory of the Discontinuity …
for r = R(ω). As a consequence of (7.31), ρ(N +1) (r) asymptotically decays more
slowly than ρ(N ) (r) (see (7.29) and (7.30)). Thus, as ω approaches zero, R(ω)
becomes infinite. For r < R(ω) and ω approaching zero, the density ρ(N ) (r) dominates the ensemble density ρ(N +ω) (r) of (7.12). Thus, in this region
lim ρ(N +ω) (r ) = ρ(N ) (r) for r < R(ω).
ω→0
(7.35)
Therefore, in the region r < R(ω), both vs(N +ω) (r) and vs(N ) (r) generate the same
density. As such these potential energies can differ at most by a constant in this
region. Since by definition both vs(N +ω) (r) and vs(N ) (r) become zero in the limit
r → ∞, we have
for r < R(ω)
vs(N +ω) (r) − vs(N ) (r) =
= 0 for r
R(ω).
(7.36)
In the limit ω → 0 the radius R(ω) becomes infinite and both potentials differ by
a constant everywhere. Employing (7.36) in the differential equation (7.20) for
the (N + ω)-electron system, and the fact that for small ω in the region r < R(ω)
the orbitals φi(N +ω) (x) ∼ φi(N ) (x), we obtain
1
(N )
− ∇ 2 + vee
(r) +
2
φi(N ) (x) =
(N +ω) (N )
φi (x)
i
for r < R(ω).
(7.37)
The corresponding equation for the N -electron system is
1
(N )
− ∇ 2 + vee
(r) φi(N ) (x) =
2
(N ) (N )
i φi (x).
(7.38)
A comparison of (7.37) and (7.38) shows that
(N +ω)
i
=
(N )
i
+
for ω → 0.
(7.39)
In particular for i = N + 1 we have
= lim
ω→0
(N +ω)
N +1
−
(N )
N +1
= −I N +1 −
(N )
N +1 ,
(7.40)
where in the last step we have used (7.33). Since
+ω)
)
v (N
(r) = v (N
H
H (r) = 0 for r → ∞,
(7.41)
7.2 Expression for Discontinuity
in Terms of S System Eigenvalues
239
we finally have
= lim
(N +ω)
(N )
vee
(r) − vee
(r)
= lim
(N +ω)
(N )
vxc
(r) − vxc
(r)
=
−
ω→0
ω→0
(N +1)
N +1
(N )
N +1 ,
(7.42)
+1)
where we have employed (N
N +1 = −I N +1 . We thus see that the discontinuity
finite. Equation (7.42) is the desired result.
is
7.3 Correlations Contributing to the Discontinuity
According To Kohn–Sham Theory
(N +ω)
If in (7.1) one employs the KS–DFT definitions of the potential energies vee
(r)
(N )
(N +ω)
(N )
KS
(r) (or vxc
(r) and vxc
(r)) as the functional derivatives δ E ee
[ρ]/δρ(r)
and vee
KS
KS
KS
[ρ]/δρ(r)| N (or δ E xc
[ρ]/δρ(r)| N +ω and δ E xc
[ρ]/δρ(r)| N ), respec| N +ω and δ E ee
tively, one is led to the conclusion that all the correlations present—Pauli, Coulomb,
and Correlation–Kinetic—contribute to the discontinuity. That this is the case may
also be surmised from (7.42), since these eigenvalues are generated via the full KS
potential energy. In earlier Q–DFT literature [10, 11], it was also implicitly assumed
that all the correlations contribute to the discontinuity. What we prove [6] via Q–DFT
in the sections to follow is that Pauli and Coulomb correlations do not contribute to the
discontinuity, and that this intrinsic property of the S system is solely a consequence
of Correlation–Kinetic effects.
7.4 Quantal Density Functional Theory of the Discontinuity
For this chapter to be self–contained, we next redefine the fields and potential energies
within Q–DFT with minor notational changes in order to distinguish between the
N - and (N + ω)-electron systems.
The N -electron Schrödinger equation and that for the corresponding S system
are, respectively,
⎡
⎣− 1
2
N
N
1
∇i2 +
v(ri ) +
2
i=1
i=1
N
i=j
⎤
1
⎦
|ri − r j |
= E (N )
(N )
(N )
(X)
(X),
(7.43)
240
and
7 Quantal Density Functional Theory of the Discontinuity …
1
(N )
− ∇ 2 + v(r) + vee
φi(N ) (x) =
2
(N ) (N )
i φi (x),
(7.44)
where (N ) (X) is the wavefunction, E the ground state energy, and φi (x) and i the
(N )
ˆ
=
single particle orbitals and eigenenergies. The density ρ(N ) (r) = (N ) |ρ(r)|
(N )
(N )
(N )
2
ˆ {φi } = σi |φi (x)| , where ρ(r)
ˆ
is the density operator (2.12),
{φi }|ρ|
and {φi(N ) } the Slater determinant of the orbitals φi(N ) (x).
(N )
(r) done to move the model fermion from a reference point at
The work vee
infinity to its position at r in the force of the conservative effective field F (N ) (r) is
where
r
F (N ) (r ) · d ,
(7.45)
(N )
)
F (N ) (r) = E (N
ee (r) + Z tc (r).
(7.46)
(N )
vee
(r) = −
∞
)
The electron–interaction component field E (N
ee (r), which is representative of Pauli
and Coulomb correlations, is obtained by Coulomb’s law from its nonlocal source
charge distribution g (N ) (rr ), the pair–correlation density. Thus,
g (N ) (rr )(r − r )
dr ,
|r − r |3
)
E (N
ee (r) =
(7.47)
ˆ
ˆ
where g (N ) (rr ) = (N ) | P(rr
)| (N ) /ρ(N ) (r), and P(rr
) is the pair–correlation
)
operator (2.28). The Correlation–Kinetic component field Z (N
tc (r) is defined in terms
(N )
(N )
of the kinetic ‘forces’ z (r; [γ]) and z s (r; [γs ]) for the interacting and S systems,
respectively, as
)
(N )
Z (N
(r; [γs ]) − z (N ) (r; [γ])
tc (r) = z s
ρ(N ) (r).
(7.48)
The nonlocal sources of the kinetic ‘forces’ are the spinless single–particle γ (N ) (rr )
and Dirac γs(N ) (rr ) density matrices, respectively, where
γ (N ) (rr ) =
(N )
|γ(rr
ˆ
)|
(N )
(7.49)
and
{φi(N ) }|γ(rr
ˆ
)| {φi(N ) }
γs(N ) (rr ) =
φi(N )∗ (rσ)φi(N ) (r σ),
=
(7.50)
σi
) is the density matrix operator (2.17). The kinetic ‘forces’ are
and where γ(rr
ˆ
defined such that the component z α(N ) (r) = 2 β ∂tαβ (r; [γ])/∂rβ , where tαβ (r) =
7.4 Quantal Density Functional Theory of the Discontinuity
241
( 41 )[∂ 2 /∂rα ∂rβ + ∂ 2 /∂rβ ∂rα ]γ (N ) (r r )|r =r =r is the kinetic–energy–density tensor.
)
The ‘force’ z (N
s (r; [γs ]) is similarly defined in terms of the S system tensor ts,αβ (r)
and Dirac density matrix γs(N ) (rr ).
Within the Schrödinger theory framework, the fractionally charged (N + ω) case
is treated in terms of an ensemble of the N - and (N + 1)-electron systems. Thus,
with the ensemble density matrix defined as in (7.11), the pair–correlation density,
and the density matrix can be shown to be
ˆ ρ(N +ω) (r)
g (N +ω) (rr ) = tr { Dˆ P}
= (1 − ω)ρ(N ) (r)g (N ) (rr )
+ ωρ(N +1) (r)g (N +1) (rr )
ρ(N +ω) (r),
(7.51)
and
γ (N +ω) (rr ) = tr { Dˆ Xˆ }
= (1 − ω)γ (N ) (rr ) + ωγ (N +1) (rr ).
(7.52)
(N +ω)
(r) in (7.23) can be rewritten as the work done in
The local potential energy vee
a conservative field F (N +ω) (r):
(N +ω)
(r) = −
vee
r
∞
F (N +ω) (r ) · d ,
with
(N +ω)
+ω)
F (N +ω) (r) = E (N
(r) + Z tc
ee
(r).
(7.53)
(7.54)
+ω)
The electron–interaction field E (N
(r) is obtained by Coulomb’s law from its
ee
source charge g (N +ω) (rr ) as
+ω)
)
(r) = (1 − ω)ρ(N ) (r)E (N
E (N
ee
ee (r)
+1)
(r)
+ ωρ(N +1) (r)E (N
ee
ρ(N +ω) (r).
(7.55)
ρ(N +ω) (r),
(7.56)
+ω)
The Correlation–Kinetic field Z(N
(r) is defined as
tc
(N +ω)
Z tc
+ω)
r; γs(N +ω)
(r) = z (N
s
− z (N +ω) r; γ (N +ω)
242
7 Quantal Density Functional Theory of the Discontinuity …
where the kinetic ‘force’ z (N +ω) (r) is obtained from its source γ (N +ω) (rr ). The
+ω)
(r) is similarly obtained from the density matrix
S system kinetic ‘force’ z (N
s
(N +ω)
constructed from the orbitals φi
(x) and is
N
γs(N +ω) (rr ) = (1 − ω)
φi(N +ω)∗ (rσ)φi(N +ω) (r σ)
σ,i=1
N +1
+ω
φi(N +ω)∗ (rσ)φi(N +ω) (r σ).
(7.57)
σ,i=1
We next prove that the discontinuity as defined by (7.1) is due to Correlation–Kinetic
effects.
7.4.1 Correlations Contributing to the Discontinuity
According To Q–DFT: Analytical Proof
A. Electron–Interaction Component
We first prove that correlations due to the Pauli exclusion principle and Coulomb
repulsion do not contribute to the discontinuity .
From (7.45) and (7.53) we have
(N +ω)
(N )
(r) − vee
(r) = − E ee (r) −
∇ vee
where
Z tc (r),
+ω)
)
(r) − E (N
E ee (r) = E (N
ee
ee (r),
and
(N +ω)
Z tc (r) = Z tc
(7.58)
(7.59)
)
(r) − Z (N
tc (r).
(7.60)
From (7.47) and (7.55), we have
E ee ([r]) =
)
(1 − ω)ρ(N ) (r) − ρ(N +ω) (r) E (N
ee (r)
+1)
(r)
+ ωρ(N +1) (r)E (N
ee
ρ(N +ω) (r).
(7.61)
Substituting for (1 − ω)ρ(N ) (r) = ρ(N +ω) (r) − ωρ(N +1) (r) into (7.61) leads to
+1)
)
(r) − E (N
E ee (r) = ωρ(N +1) (r) E (N
ee
ee (r)
ρ(N +ω) (r).
(7.62)
It follows from (7.62) that limω→0 E ee (r) = 0. To see this consider the radius
R(ω) defined by (7.34). For r < R(ω) and small ω, the density ρ(N ) (r) dominates the
ensemble density ρ(N +ω) (r). Thus, in this region ρ(N +ω) (r) ∼ ρ(N ) (r), and E ee (r) is
7.4 Quantal Density Functional Theory of the Discontinuity
243
linear in ω, and vanishes as ω → 0. For r
R(ω), the ensemble density ρ(N +ω) (r) ∼
(N +1)
(r). Substitution into (7.62) shows that the ω’s cancel. But in this region the
ωρ
+1)
)
2
(r) − E (N
difference [E (N
ee
ee (r)] ∼ 1/r so that E ee (r) once again vanishes. In the
region r ∼ R(ω), E ee (r) vanishes essentially linearly with ω. We note, however,
that E ee (r) is finite for positive definite ω, irrespective of how small ω is. It is only
in the limit of vanishing ω that the Pauli and Coulomb correlation contributions to
the discontinuity vanish.
Finally, since the pair-correlation density may be written as g (N ) (rr ) = ρ(N ) (r ) +
(N )
)
ρxc (rr ), where ρ(N
xc (rr ) is the Fermi–Coulomb hole charge distribution, we
+ω)
)
have E ee (r) = E H (r) + Exc (r), where E H (r) = [E (N
(r) − E (N
H
H (r)] and
(N )
+ω)
)
)
(r) − E (N
and E (N
Exc (r) = [E (N
xc
xc (r)]. Here E H
xc (r) are the Hartree and
Pauli–Coulomb fields arising from the component charge distributions ρ(N ) (r )
(N +1)
)
)
(r) − E (N
and ρ(N
xc (rr ), respectively. Since E H (r) = ω[E H
H (r)], it follows that
limω→0 E H (r) = 0, and, consequently, the limω→0 E xc (r) = 0.
B. Correlation–Kinetic Component
Since the quantum–mechanical electron–interaction contribution
vanishes in the lim ω → 0, we have
(N +ω)
(N )
(r) − vee
(r) = − Z tc (r),
lim ∇ vee
ω→0
E ee (r) in (7.58)
(7.63)
which proves the fact that the discontinuity is strictly a Correlation–Kinetic effect.
The discontinuity is then the work done
=−
0
∞
(N +ω)
Z tc
)
(r ) − Z (N
tc (r ) · d .
(7.64)
From (7.63) it also follows that this work done is path–independent. Equation (7.64)
is an alternate expression for the discontinuity , in which it is evident that the
correlations that contribute to it are solely those due to Correlation–Kinetic effects.
To understand more fundamentally how Correlation–Kinetic effects contribute to
the discontinuity, we next explain the structure of Z tc (r) for small ω. We rewrite
Z tc (r) as
(N +ω)
Z tc (r) = Z tc
)
(r) − Z (N
tc (r)
= A + B,
(7.65)
(7.66)
where
A=
1
1
)
z (N +ω) (r) − (N ) z (N
s (r)
ρ(N +ω) (r) s
ρ
(7.67)
244
7 Quantal Density Functional Theory of the Discontinuity …
and
B=−
+
1
ρ(N +ω) (r)
(1 − ω)z (N ) (r) + ωz (N +1) (r)
1
z (N ) (r).
ρ(N ) (r)
(7.68)
dominates,
γs(N +ω)
For
r < R(ω),
the
region
where
ρ(N ) (r)
(N )∗
(N )
N
(N +ω)
)
(rr ) ∼ σ,i=1 φi (rσ)φi (r σ), so that z s
(r) ∼ z (N
s (r). Therefore, A = 0.
The term B = 0, since the terms linear in ω are negligible. Thus, in this region,
Z tc (r) = 0.
(N +ω)
)
(r) and Z (N
In the r → ∞ limit, both Z tc
tc (r) vanish, so that in this region
Z tc (r) = 0.
For r
R(ω), we have φi(N +ω) (r) → φi(N +1) (r), so that γs(N +ω) (rr )
N +1
+ω)
+1)
∼ ω σ,i=1
φi(N +1)∗ (rσ)φi(N +1) (r σ) and z (N
(r) ∼ ωz (N
(r). Thus A ∼
s
s
(N +1)
(N +1)
(N )
(N )
(N +1)
(r)/ρ
(r)] − [z s (r)/ρ (r)]} and B ∼ {[−z
(r)/ρ(N +1) (r)] +
{[z s
(N
+1)
(N
)
[z (N ) (r)/ρ(N ) (r)]}, so that Z tc (r) = Z tc
(r) − Z tc (r). Thus, in this region,
Ztc (r) is finite. In this limit, as ω → 0, the radius R(ω) becomes infinite, and the
difference Z tc (r) stabilizes. We next demonstrate the above conclusions via two
numerical examples.
7.4.2 Numerical Examples
Example 1. As a demonstration of the above conclusions, we consider the
example where the integer system is the H e+ ion (atomic number Z = 2, electron
number N = 1). Its wavefunction, which is hydrogenic, is known, as is the density
ρ(N =1) (r) ≡ ρ(1) (r). The ensemble density ρ(1+ω) (r) of (7.12) is then
ρ(1+ω) (r) = (1 − ω)ρ(1) (r) + ωρ(2) (r),
(7.69)
where ρ(2) (r) ≡ ρ(N +1) (r) is the density of the H e atom. For the H e atom, a highly
accurate 491–parameter correlated wavefunction [12] is employed. This wavefunction is accurate upto r = 24 a.u. from the nucleus, which in essence is infinity for
the atom. Smaller and smaller fractional charge ω is added to the ls shell of the H e+
ion. For the corresponding S system, there is therefore only one orbital φ(1+ω) (r).
Thus, the ensemble density ρ(1+ω) (r) in terms of the S system orbitals as given by
(7.22) is
ρ(1+ω) (r) = (1 − ω)|φ(1+ω) (r)|2 + ω2|φ(1+ω) (r)|2
= (1 + ω)|φ(1+ω) (r)|2 .
(7.70)
7.4 Quantal Density Functional Theory of the Discontinuity
245
Fig. 7.2 (a) Hartree field
difference E H (r) for
different fractional charge ω.
The integer charge system is
H e+ . (b) Work done
W H (r) in the field E H
Thus, from (7.69) and (7.70), we have
φ(1+ω) (r) =
(1 − ω)ρ(1) (r) + ωρ(2) (r)
1+ω
1/2
.
(7.71)
As the wavefunctions for H e+ and H e, and consequently the orbital φ(1+ω) (r), are
known, all the requisite sources and fields can then be determined for different values
of the fractional charge ω.
In Fig. 7.2(a) we plot the Hartree field difference E H (r) for ω = 10−1 , 10−2 ,
and 10−3 . As the fractional charge diminishes, the difference E H (r) becomes negr
ligible. The corresponding work W H (r) = − ∞ E H (r ) · d is constant in the
interior as expected (Fig. 7.2(b)), but becomes smaller with decreasing fractional
charge, although it is still finite at ω = 10−3 . As ω is decreased further, however,
both E H (r) and W H (r) vanish. Thus the Hartree component of the Coulomb
interaction does not contribute to the discontinuity.
In Fig. 7.3(a), the difference in the Pauli–Coulomb fields Exc (r) is plotted
for fractional charges ω = 10−5 , 10−8 , and 10−10 . As expected, it vanishes in the
246
7 Quantal Density Functional Theory of the Discontinuity …
Fig. 7.3 (a) Pauli-Coulomb
field difference Exc (r) for
different fractional charge ω.
The integer charge system is
H e+ . (b) Work done
Wxc (r) in the field
Exc (r)
interior, and is peaked in the surface region. It diminishes with decreasing ω, while
simultaneously the peak moves further into the classically forbidden region where
Exc (r) decays as (1 − ω)/r 2 for finite ω. Thus, the corresponding work (Fig. 7.3(b))
r
Wxc (r) = − ∞ E xc (r ) · d is constant in the interior, with the region where
this difference is constant increasing with decreasing ω. Furthermore, as expected
the constant value of Wxc (r) also diminishes with decreasing ω. For ω = 10−10
the constant value of Wxc (r) in the interior is 0.052 a.u., Asymptotically, Wxc (r)
decays as (1 − ω)/r . With vanishing fractional charge, the Pauli–Coulomb contribution to the discontinuity will also vanish.
In Fig. 7.4 we plot the difference Z tc (r) of the Correlation–Kinetic fields, for
fractional charges ω = 10−2 , 10−5 , 10−8 , and 10−10 a.u., As expected, this difference vanishes in the interior region. However, in the surface region, these curves
are dramatically different from those of Figs. 7.2 and 7.3 in that as the fractional
charge ω is decreased, the magnitude of these curves increases (Fig. 7.4(a)). With
a further decrease in ω, the structure essentially stabilized (Fig. 7.4(b)) and remains
finite, while simultaneously moving further out into the classically forbidden region.
r
Thus the constant value of the work Wtc (r) = − ∞ Z tc (r ) · d increases with
7.4 Quantal Density Functional Theory of the Discontinuity
247
Fig. 7.4 (a, b) Correlation
Kinetic field difference
Ztc (r) for difference
fractional charge ω. The
integer charge system is
H e+
decreasing fractional charge (Fig. 7.5), approaching the exact value of the discontinuity
from below. This value may be determined from the known result
+1)
(N )
−
= (N
N +1
N +1 of (7.42), and the fact the highest occupied eigenvalue of the
S system corresponds to minus the ionization potential. Taking into consideration
the double occupancy of the 1s orbital, and that the ionization potentials of H e and
H e+ are 0.903 and 2 a.u., respectively, we have = 1.097 a.u., The value of Wtc
for ω = 10−10 is 1.035 a.u., Adding the value of Wxc = 0.052 a.u. for the same ω
value, we obtain = 1.087 a.u., which is essentially exact. In the limit of vanishing
ω, the contribution from Wxc will vanish, and that due to Wtc will equal . This
confirms that the discontinuity in the electron–interaction potential energy is solely
due to Correlation–Kinetic effects.
Example 2. The calculations in this second example are performed within the
Pauli–correlated approximation of Q–DFT [13] as described in Sect. 5.8.1. (see
Chap. 6 of QDFT2) (This is also the lowest–order of Q–DFT many–body perturbation theory. See Chap. 18 of QDFT2.) In this approximation, only correlations
due to the Pauli exclusion principle are considered beyond the Hartree term. Thus,
the corresponding pair–correlation density gs(N ) (rr ) is determined from a Slater